It depends on what you're playing. When we play LARP in Russian, we typically use the word хиты (sing. хит). When playing computer games, we transliterate HP as ХП (read as [хэпэ]). You may see expressions like полХП ("Half of the HP bar"), треть ХП etc. It isn't uncommon to translate "HP" (Health Points) directly as Очки здоровья, abbreviated to ОЗ.
If you use the "wrong" word, for example, say "У меня мало хитов осталось" while playing a computer game, or "Мало [хэпэ]" when playing LARP, you'll sound... strange? :) I've actually never heard ОЗ from any kind of gamer.
Keep in mind that some systems use different units to measure damage. For example, Vampire: The Masquerade tabletop uses "health levels" that are pretty much the same for all humanoid characters. It would be wrong to translate that as either хиты, ОЗ or ХП; translate it directly as "уровни здоровья", or, if you want, as "уровни ранений" etc.
Using the word "life" (жизнь) isn't really common, since games that made you lose one of several lives for making a mistake stayed somewhere in the 80s, most games don't use that concept. I'd expect an expression like that from a kid (because kids' language is a bit different from adult language, and is a bit stuck in time), or from a mom who's also stuck.
Awesome question, mate, thank you for that. If you need any clarification, ask in the comments.